It is known that for a complex manifold, coordinates exist such that the complex structure takes the canonical form $$ J=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array}\right] $$ in a local patch. These coordinates are known as local holomorphic coordinates. Is there an analogous concept for quaternionic manifolds? I.e., are there special coordinates whereby the quaternionic structure is defined by the canonical form
$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$
$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right]? $$
Briefly, "no": The quaternionic analogues of the Cauchy-Riemann equations (I've heard them called Cauchy-Feuter equations, see also MO Does Feuter regularity imply derivability in all directions?) imply affineness, so unless the manifold admits quaternionic charts whose overlap maps are affine (e.g., tori) there's no intrinsic notion of "quaternionic coordinates". (It's not the coordinates themselves that determine "holomorphicity", incidentally, it's the overlap maps. This point is often misunderstood.)
In case it's of interest, there are quaternionic and hyper-Kähler manifolds. Hyper-Kähler manifolds seem likely to be the differential-geometric structure you're looking for.